Theorem int0 4926

Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) (Proof shortened by JJ, 26-Jul-2021.)

Assertion

Ref	Expression
int0	⊢ ∩ ∅ = V

Proof of Theorem int0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4458	. . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥
2		vex 3457	. . . . 5 ⊢ 𝑦 ∈ V
3	2	elint2 4918	. . . 4 ⊢ (𝑦 ∈ ∩ ∅ ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥)
4	1, 3	mpbir 234	. . 3 ⊢ 𝑦 ∈ ∩ ∅
5	4, 2	2th 267	. 2 ⊢ (𝑦 ∈ ∩ ∅ ↔ 𝑦 ∈ V)
6	5	eqriv 2758	1 ⊢ ∩ ∅ = V

Syntax hints:   = wceq 1568   ∈ wcel 2141  ∀wral 3077  Vcvv 3453  ∅c0 4285  ∩ cint 4911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-v 3455  df-dif 3907  df-nul 4286  df-int 4912

This theorem is referenced by:  unissint  4936  uniintsn  4949  rint0  4952  intex  5314  intnex  5315  oev2  8507  fiint  9285  elfi2  9373  fi0  9379  cardmin2  9984  00lsp  21081  cmpfi  23544  ptbasfi  23717  fbssint  23974  fclscmp  24166  zarcmplem  34237  rankeq1o  36629  bj-0int  37709  heibor1lem  38426  ipoglb0  49739  mreclat  49742